naginterfaces.library.rand.times_​garch_​exp

naginterfaces.library.rand.times_garch_exp(dist, num, ip, iq, theta, df, fcall, comm, statecomm)

times_garch_exp generates a given number of terms of an exponential $\text{GARCH}(p,q)$ process (see Engle and Ng (1993)).

For full information please refer to the NAG Library document for g05pg

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g05/g05pgf.html

Parameters

diststr, length 1

The type of distribution to use for ${ϵ}_t$.

$dist=\text{'N'}$

A Normal distribution is used.

$dist=\text{'T'}$

A Student’s $t$-distribution is used.

numint

$T$, the number of terms in the sequence.

ipint

The number of coefficients, ${\beta }_{\text{i}}$, for $\text{i}=1,2,\dots ,p$.

iqint

The number of coefficients, ${\alpha }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$.

thetafloat, array-like, shape $(2\times iq+ip+1)$

The initial parameter estimates for the vector $\theta$. The first element must contain the coefficient ${\alpha }_o$ and the next $iq$ elements must contain the autoregressive coefficients ${\alpha }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$. The next $iq$ elements must contain the coefficients ${\varphi }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$. The next $ip$ elements must contain the moving average coefficients ${\beta }_{\text{j}}$, for $\text{j}=1,2,\dots ,p$.

dfint

The number of degrees of freedom for the Student’s $t$-distribution.

If $dist=\text{'N'}$, $df$ is not referenced.

fcallbool

If $fcall=True$, a new sequence is to be generated, otherwise a given sequence is to be continued using the information in $comm$[‘r’].

commdict, communication object, modified in place

Communication structure for the reference vector.

If $fcall=False$, this argument must have been initialized by a prior call to times_garch_exp.

statecommdict, RNG communication object, modified in place

RNG communication structure.

This argument must have been initialized by a prior call to init_repeat() or init_nonrepeat().

Returns

htfloat, ndarray, shape $\left(num\right)$

The conditional variances $h_{\text{t}}$, for $\text{t}=1,2,\dots ,T$, for the $\text{GARCH}(p,q)$ sequence.

etfloat, ndarray, shape $\left(num\right)$

The observations ${ϵ}_{\text{t}}$, for $\text{t}=1,2,\dots ,T$, for the $\text{GARCH}(p,q)$ sequence.

Raises

NagValueError

(errno $1$)

On entry, $dist$ is not valid: $dist=⟨value⟩$.

(errno $2$)

On entry, $num=⟨value⟩$.

Constraint: $num\ge 0$.

(errno $3$)

On entry, $ip=⟨value⟩$.

Constraint: $ip\ge 0$.

(errno $4$)

On entry, $iq=⟨value⟩$.

Constraint: $iq\ge 1$.

(errno $6$)

On entry, $df=⟨value⟩$.

Constraint: $df\ge 3$.

(errno $10$)

$ip$ or $iq$ is not the same as when $comm$[‘r’] was set up in a previous call.

Previous value of $ip=⟨value⟩$ and $ip=⟨value⟩$.

Previous value of $iq=⟨value⟩$ and $iq=⟨value⟩$.

(errno $12$)

On entry, $statecomm$[‘state’] vector has been corrupted or not initialized.

(errno $20$)

Invalid sequence generated, use different parameters.

Notes

An exponential $\text{GARCH}(p,q)$ process is represented by:

$$ln\left(h_t\right)={\alpha }_0+\sum _{i=1}^q{\alpha }_i{z}_{t-i}+\sum _{i=1}^q{\varphi }_i(\left|z_{t-i}\right|-E\left[\left|z_{t-i}\right|\right])+\sum _{j=1}^p{\beta }_{j}ln\left(h_{t-j}\right)\text{,}t=1,2,\dots ,T\text{;}$$

where $z_t=\frac{{ϵ}_t}{\sqrt{h_t}}$, $E\left[\left|z_{t-i}\right|\right]$ denotes the expected value of $\left|z_{t-i}\right|$, and ${ϵ}_t|{\psi }_{t-1}=N(0,h_t)$ or ${ϵ}_t|{\psi }_{t-1}=S_t(\text{df},h_t)$. Here $S_t$ is a standardized Student’s $t$-distribution with $\text{df}$ degrees of freedom and variance $h_t$, $T$ is the number of observations in the sequence, ${ϵ}_t$ is the observed value of the $\text{GARCH}(p,q)$ process at time $t$, $h_t$ is the conditional variance at time $t$, and ${\psi }_t$ the set of all information up to time $t$.

One of the initialization functions init_repeat() (for a repeatable sequence if computed sequentially) or init_nonrepeat() (for a non-repeatable sequence) must be called prior to the first call to times_garch_exp.

References

Bollerslev, T, 1986, Generalised autoregressive conditional heteroskedasticity, Journal of Econometrics (31), 307–327

Engle, R, 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica (50), 987–1008

Engle, R and Ng, V, 1993, Measuring and testing the impact of news on volatility, Journal of Finance (48), 1749–1777

Glosten, L, Jagannathan, R and Runkle, D, 1993, Relationship between the expected value and the volatility of nominal excess return on stocks, Journal of Finance (48), 1779–1801

Hamilton, J, 1994, Time Series Analysis, Princeton University Press